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"Illicit" can be found in any English dictionary. "The YouTube variety" refers to the readily available pornography found on YouTube. Questions on this site are expected to show that the questioner has done a modicum of research before posting.
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John M. LandsbergAug 15 '13 at 5:03

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This question would appear to be more suitable for our sister site, English Language Learners, since it is about simple word definitions. Have you tried looking up the words you find difficult in the dictionary? If so, please indicate what research you have done yourself; otherwise, please look them up and ask more specifically what you find difficult about the sentence.
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Janus Bahs JacquetAug 15 '13 at 5:03

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Three miles away Patrick Beeman would be working down a list of bullet points with Lorna: General Procedure, Custody, Money, House, Pensions, Miscellaneous, Standard Fees. My mother is going to crumple up: we've kept The Troubles well hidden. "Whose idea was it first?" she'll want to know, meaning, "Whose fault is it?" There's no Somebody Else on Lorna's side, I'm sure – she's too miserable to be in love – and God knows I've been too busy trying to keep my alleged ­ consultancy afloat to think about illicit sex, beyond the YouTube variety. Though sure, there's no denying that the money stuff hasn't helped the marital stuff.

This is someone contemplating marital difficulties (The Troubles1). Sex outside the marriage ("illicit sex") is not part of the cause, because his wife Lorna is too miserable and he is too busy. Worrying about money and the need to keep what should be a consultancy business in business is contributing to the difficulty, though.

The "YouTube variety" of sex is online pornography, which presumably he has been partaking of, probably in lieu of the real thing.

1 The conflict in Northern Ireland (1969–1998) was called The Troubles. Mitchell is indulging in black humour in comparing the marriage to that situation.

Do you know there are ice-cream flavours beyond those found on the shelves?

I am reading news of illicit affairs beyond the variety found in the Variety magazine.

Perhaps, this is a scenario which could result in someone making that statement:

I have a business to run. Some people don't think I could even call it a consultancy. Allegedly, it might not even qualify as a consultancy. I am too busy keeping that business afloat. Sometimes I take a break watching youtube. I do find occasional incidences of illicit sex on youtube. I am not interested in looking for videos on illicit sex. I don't have time to pay attention to videos of illicit sex, beyond the various types that I occasionally encounter on youtube.